$f(x, y, z) = xz^2 + e^x - yz$ What are all the critical points of $f$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(0, 0, 0)$ (Choice B) B $(0, 1, -1)$ (Choice C) C $(0, -1, -1)$ (Choice D) D There are no critical points.
A critical point of a scalar field $f$ is where $\nabla f = \bold{0}$. [What's that bolded 0?] Let's find the gradient of $f$ ! $\nabla f = \begin{bmatrix} z^2 + e^x \\ \\ -z \\ \\ 2xz - y \end{bmatrix}$ Notice that the $x$ -component of the gradient is always greater than $0$. In other words, no matter what input we feed the gradient of $f$, it will never equal the zero vector. Therefore, there are no critical points.